Normal Hypothesis Testing (AQA A Level Maths: Statistics)

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Normal Hypothesis Testing

How is a hypothesis test carried out with the normal distribution?

  • The population parameter being tested will be the population mean, in a normally distributed random variable begin mathsize 16px style N left parenthesis mu comma sigma squared right parenthesis end style
    • The population mean is tested by looking at the mean of a sample taken from the population
    • The sample mean is denoted begin mathsize 16px style x with bar on top end style
    • For a random variable X tilde size 16px N size 16px left parenthesis size 16px mu size 16px comma size 16px sigma to the power of size 16px 2 size 16px right parenthesis the distribution of the sample mean would be size 16px X with size 16px bar on top tilde size 16px N size 16px left parenthesis size 16px mu size 16px comma size 16px sigma to the power of size 16px 2 over size 16px n size 16px right parenthesis
  • A hypothesis test is used when the value of the assumed population mean is questioned
  • The null hypothesis, H0 and alternative hypothesis, H1 will always be given in terms of µ
    • Make sure you clearly define µ before writing the hypotheses, if it has not been defined in the question
    • The null hypothesis will always be H0 : µ = ...
    • The alternative hypothesis will depend on if it is a one-tailed or two-tailed test
    • A one-tailed test would test to see if the value of  µ has either increased or decreased
      • The alternative hypothesis, H1 will be H1 :  µ > ... or H1 :  µ < ...
    • A two-tailed test would test to see if the value of µ has changed
      • The alternative hypothesis, H1 will be H1 :  µ ≠ ..
  • To carry out a hypothesis test with the normal distribution, the test statistic will be the sample mean, bold italic X with bar on top
    • Remember that the variance of the sample mean distribution will be the variance of the population distribution divided by n
    • the mean of the sample mean distribution will be the same as the mean of the population distribution
  • The normal distribution will be used to calculate the probability of the observed value of the test statistic taking the observed value or a more extreme value
  • The hypothesis test can be carried out by
    • either calculating the probability of the test statistic taking the observed or a more extreme value (p – value) and comparing this with the significance level
    • or by finding the critical region and seeing whether the observed value of the test statistic lies within it
      • Finding the critical region can be more useful for considering more than one observed value or for further testing

How is the critical value found in a hypothesis test for the mean of a normal distribution?

  • The critical value(s) will be the boundary of the critical region
    • The probability of the observed value being within the critical region, given a true null hypothesis will be the same as the significance level
  • For an  alpha%  significance level:
    • In a one-tailed test the critical region will consist of  alpha% in the tail that is being tested for
    • In a two-tailed test the critical region will consist of alpha over 2 percent sign in each tail
  • To find the critical value(s) find the distribution of the sample means, assuming H0 is true, and use the inverse normal function on your calculator
  • For a two-tailed test you will need to find both critical values, one at each end of the distribution

What steps should I follow when carrying out a hypothesis test for the mean of a normal distribution?

  • Following these steps will help when carrying out a hypothesis test for the mean of a normal distribution:

Step 1.  Define the distribution of the population mean usually X tilde size 16px N size 16px left parenthesis size 16px mu size 16px comma size 16px sigma to the power of size 16px 2 size 16px right parenthesis

Step 2.  Write the null and alternative hypotheses clearly using the form

H0 : μ = ...

H1 : μ ... ...

Step 3.   Assuming the null hypothesis to be true, define the test statistic, usually size 16px X with size 16px bar on top tilde size 16px N size 16px left parenthesis size 16px mu size 16px comma size 16px sigma to the power of size 16px 2 over size 16px n size 16px right parenthesis

Step 4.   Calculate either the critical value(s) or the p – value (probability of the observed value) for the test

Step 5.   Compare the observed value of the test statistic with the critical value(s) or the p - value with the significance level

Step 6.   Decide whether there is enough evidence to reject H0 or whether it has to be accepted

Step 7.  Write a conclusion in context

Worked example

The time, X  minutes, that it takes Amelea to complete a 1000-piece puzzle can be modelled using X tilde straight N left parenthesis 204 comma 81 right parenthesis .  Amelea gets prescribed a new pair of glasses and claims that the time it takes her to complete a 1000-piece puzzle has decreased.  Wearing her new glasses, Amelea completes 12 separate 1000-piece puzzle and calculates her mean time on these puzzles to be 201 minutes.  Use these 12 puzzles as a sample to test, at the 5% level of significance, whether there is evidence to support Amelea’s claim. You may assume the variance is unchanged.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.